Question

Find the charge per unit length of inner surface and outer surface of the outer cylinder :

• A. $-\lambda$ on inner, $+\lambda$ on outer surface
• B. $-\lambda$ on outer, $+\lambda$ on inner surface
• C. $-\lambda /2$ on both inner and outer surfaces
• D. $+\lambda /2$ on both inner and outer surfaces

Answer 1

The correct option is A $-\lambda$ on inner, $+\lambda$ on outer surface

Consider a Gaussian cylinder of length $l$ and radius $r(b<r<c)$ as shown in the figure.

The curved surface of the cylinder lies within the conducting cylinder.

Since electric field inside the conductor is zero.

Hence $\overrightarrow{E}=\overrightarrow{0}$

Thus, flux through the curved surface is zero.

As the electric field is radial, the flux through the lids of the cylinder is also zero.

By Gauss' law, $\oint E.ds=\frac{q_{enclosed}}{{ϵ}_o}$

Thus the total charge enclosed in the cylindrical surface is zero.

But, the inner cylindrical conductor carries a charge of $\lambda l$.

Hence, the inner surface of outer cylinder must carry a charge of $-\lambda l$.

Linear charge density on the inner surface = $\frac{-\lambda l}{l}=-\lambda$

Initially, the charge on outer cylinder is zero.

Hence by principle of conservation of charge, total charge on outer surface and inner surface is zero.

Thus, the linear charge density on the outer surface is $+\lambda$.